# Import required libraries
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns

#Data loading
data = pd.read_csv('hw1_input.csv')
data

#Check summary statistics
data.describe()

#Check correlation
data.corr()

sns.pairplot(data)

/Users/eren/micromamba/envs/Py_ML_DS_env/lib/python3.9/site-packages/seaborn/axisgrid.py:64: UserWarning: The figure layout has changed to tight
  self.fig.tight_layout(*args, **kwargs)

<seaborn.axisgrid.PairGrid at 0x28172ac70>

#Visualisation for correlation
sns.heatmap(data.corr())

<Axes: >

sns.scatterplot(data['height of substrate'], data['width of patch'])

/Users/eren/micromamba/envs/Py_ML_DS_env/lib/python3.9/site-packages/seaborn/_decorators.py:36: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.
  warnings.warn(

<Axes: xlabel='height of substrate', ylabel='width of patch'>

# Load the data
real_S11 = pd.read_csv('hw1_real.csv', header=None)  # Real part of S11
imag_S11 = pd.read_csv('hw1_img.csv', header=None)   # Imaginary part of S11

real_S11[1::].reset_index(drop=True)

df = pd.merge(data, real_S11[1::].reset_index(drop=True)[[100]], left_index=True, right_index=True)
df

# Relationships between the features and the real part at a single frequency point
sns.pairplot(df, y_vars=[100])

/Users/eren/micromamba/envs/Py_ML_DS_env/lib/python3.9/site-packages/seaborn/axisgrid.py:64: UserWarning: The figure layout has changed to tight
  self.fig.tight_layout(*args, **kwargs)

<seaborn.axisgrid.PairGrid at 0x2d5d443d0>

df.corr()[100]

length of patch                              -0.027677
width of patch                                0.830605
height of patch                               0.071386
height of substrate                           0.880056
height of solder resist layer                -0.007944
radius of the probe                          -0.035128
c_pad                                        -0.040812
c_antipad                                    -0.028853
c_probe                                       0.110294
dielectric constant of substrate              0.377078
dielectric constant of solder resist layer    0.033185
100                                           1.000000
Name: 100, dtype: float64

# Relationships between the target (frequency 100) and width of patch feature
plt.scatter(df['width of patch'], df[100])

<matplotlib.collections.PathCollection at 0x33ce52760>

# Calculate the magnitude of S11
S11_magnitude = np.sqrt(real_S11**2 + imag_S11**2)

S11_target=S11_magnitude[1::].reset_index(drop=True)
S11_target

df = pd.merge(data, S11_target[[100]], left_index=True, right_index=True)
df

# Relationships between the features and the magnitude at a single frequency point
sns.pairplot(df, y_vars=[100])

/Users/eren/micromamba/envs/Py_ML_DS_env/lib/python3.9/site-packages/seaborn/axisgrid.py:64: UserWarning: The figure layout has changed to tight
  self.fig.tight_layout(*args, **kwargs)

<seaborn.axisgrid.PairGrid at 0x2d62b3d60>

# Plot the magnitude for the first 12 designs
fig, ax = plt.subplots(figsize=(10, 6))
# Customize the plot
ax.set_title('S11 Magnitudes for the First 12 Designs Across Frequency Points')
ax.set_xlabel('Frequency Index')
ax.set_ylabel('S11 Magnitude')
ax.grid(True)
# Add a caption below the plot
plt.figtext(0.5, -0.05, "S11 magnitude values showing antenna reflection performance over 201 frequency points and the absolute minimum points for each design where it has the strongest reflection performance. ", 
            ha="center", fontsize=10, wrap=True)

for i in range(1, 13):  # First 12 designs
    plt.plot(S11_magnitude.iloc[i, :], label=f"Design {i}")
    # Add labels to specific points
    min_value = np.min(S11_magnitude.iloc[i, :])
    min_index = np.argmin(S11_magnitude.iloc[i, :])
    plt.text(min_index, min_value, f'x ({min_index}, {min_value:.5f})', fontsize=8, ha='left')
ax.legend(loc="best", fontsize=10)

<matplotlib.legend.Legend at 0x369972a90>

# Find the minimum value and its index for the 3th design
min_value = np.min(S11_magnitude.iloc[3, :])
min_index = np.argmin(S11_magnitude.iloc[3, :])
print(f"Minimum value: {min_value} at index {min_index}")

Minimum value: 0.02148968319005195 at index 67

# Plot the magnitude for the first 6 designs
fig, ax = plt.subplots(figsize=(10, 6))
# Customize the plot
ax.set_title('3th Design')
ax.set_xlabel('Frequency Index')
ax.grid(True)
# Add a caption below the plot
plt.figtext(0.5, -0.05, "For example, the green design shows the S11 value approaching zero (0.02148968319005195) around the frequency at index 70 (67). Smaller magnitudes indicate less signal loss, meaning a stronger reflection performance. In the case of S11, a smaller magnitude implies that the signal can be received more effectively around that frequency.", 
            ha="center", fontsize=10, wrap=True)
min_value = np.min(S11_magnitude.iloc[3, :])
min_index = np.argmin(S11_magnitude.iloc[3, :])
plt.text(min_index, min_value, f'x ({min_index}, {min_value:.5f})', fontsize=8, ha='left')
plt.plot(S11_magnitude.iloc[3, :], label="Magnitude", c='g', lw=2.5)
plt.plot(real_S11.iloc[3, :], label="Real Part", c='r',ls=':', lw=1.5)
plt.plot(imag_S11.iloc[3, :], label="Imaginary Part", c='b',ls=':',lw=1.5)

ax.legend(loc="best", fontsize=10)

<matplotlib.legend.Legend at 0x297579c70>

indices=[]
# Plot all designs' critical frequencies where antenna's performance is the best
fig, ax = plt.subplots(figsize=(10, 6))
# Customize the plot
ax.set_title('Frequency Values and S11 Magnitudes For Each Design Where It Has The Strongest Reflection Performance.')
ax.set_xlabel('Frequency Index')
ax.set_ylabel('S11 Magnitude')
ax.grid(True)
# Add a caption below the plot
plt.figtext(0.5, -0.05, "Showing the frequency points for each design where it has the strongest reflection performance. ", 
            ha="center", fontsize=10, wrap=True)
for i in range(1, len(S11_magnitude)): 
    # Add labels to specific points
    min_value = np.min(S11_magnitude.iloc[i, :])
    min_index = np.argmin(S11_magnitude.iloc[i, :])
    plt.scatter(min_index, min_value, c='b', s=0.9)
    plt.text(min_index, min_value, f'x ({min_index}, {min_value:.5f})', fontsize=3, ha='left')
    indices.append(min_index)

fig, ax = plt.subplots(figsize=(10, 6))
ax.set_xlabel('Frequency Index')
ax.set_ylabel('Counts')
plt.hist(indices, bins=40);

# Omit the frequency points 0-10 and 190-200
fig, ax = plt.subplots(figsize=(10, 6))
ax.set_xlabel('Frequency Index')
ax.set_ylabel('Counts')
plt.hist(indices, bins=10, range=(10, 190));

# Replace with actual indices based on your analysis
key_frequencies = [0, 100, 200]

values=[]
values_real=[]
values_imag=[]
for i in range(0, len(S11_magnitude)): 
    # Add labels to specific points
    min_value = np.min(S11_magnitude.iloc[i, :])
    values.append(min_value)
    min_value = np.min(real_S11[1::].reset_index(drop=True).iloc[i, :])
    values_real.append(min_value)
    min_value = np.min(imag_S11[1::].reset_index(drop=True).iloc[i, :])
    values_imag.append(min_value)

values_df=pd.DataFrame(values, columns=["resonance"])

values_real_df=pd.DataFrame(values_real, columns=["resonance_real"])

values_imag_df=pd.DataFrame(values_imag, columns=["resonance_imag"])

#### Adding Resonance Values As The Target
df = pd.merge(data, values_df , left_index=True, right_index=True)
df = pd.merge(df, values_real_df , left_index=True, right_index=True)
df = pd.merge(df, values_imag_df , left_index=True, right_index=True)
df

df.corr()['resonance']

length of patch                               0.088859
width of patch                               -0.719391
height of patch                              -0.057423
height of substrate                          -0.727314
height of solder resist layer                 0.006547
radius of the probe                           0.016414
c_pad                                         0.076387
c_antipad                                     0.018615
c_probe                                       0.044486
dielectric constant of substrate             -0.498250
dielectric constant of solder resist layer   -0.107778
resonance                                     1.000000
resonance_real                               -0.698963
resonance_imag                                0.411778
Name: resonance, dtype: float64

# Relationships between the features and the resonance values
sns.pairplot(df, y_vars=['resonance'])

/Users/eren/micromamba/envs/Py_ML_DS_env/lib/python3.9/site-packages/seaborn/axisgrid.py:64: UserWarning: The figure layout has changed to tight
  self.fig.tight_layout(*args, **kwargs)

<seaborn.axisgrid.PairGrid at 0x2d98ee670>

df.corr()['resonance_real']

length of patch                               0.009014
width of patch                                0.894428
height of patch                               0.074876
height of substrate                           0.960081
height of solder resist layer                 0.004733
radius of the probe                          -0.049564
c_pad                                        -0.028054
c_antipad                                     0.000707
c_probe                                       0.007350
dielectric constant of substrate              0.401108
dielectric constant of solder resist layer    0.012930
resonance                                    -0.698963
resonance_real                                1.000000
resonance_imag                               -0.187013
Name: resonance_real, dtype: float64

# Relationships between the features and the resonance values (real part)
sns.pairplot(df, y_vars=['resonance_real'])

/Users/eren/micromamba/envs/Py_ML_DS_env/lib/python3.9/site-packages/seaborn/axisgrid.py:64: UserWarning: The figure layout has changed to tight
  self.fig.tight_layout(*args, **kwargs)

<seaborn.axisgrid.PairGrid at 0x2d98dd4c0>

df.corr()['resonance_imag']

length of patch                               0.057338
width of patch                               -0.197444
height of patch                               0.065604
height of substrate                          -0.267361
height of solder resist layer                 0.023461
radius of the probe                          -0.139563
c_pad                                         0.081723
c_antipad                                     0.152817
c_probe                                      -0.391245
dielectric constant of substrate             -0.203338
dielectric constant of solder resist layer   -0.009696
resonance                                     0.411778
resonance_real                               -0.187013
resonance_imag                                1.000000
Name: resonance_imag, dtype: float64

# Relationships between the features and the resonance values (imaginary part)
sns.pairplot(df, y_vars=['resonance_imag'])

/Users/eren/micromamba/envs/Py_ML_DS_env/lib/python3.9/site-packages/seaborn/axisgrid.py:64: UserWarning: The figure layout has changed to tight
  self.fig.tight_layout(*args, **kwargs)

<seaborn.axisgrid.PairGrid at 0x2d9f05d90>

## Scikit-Learn Implementations
from sklearn.preprocessing import StandardScaler
from sklearn.decomposition import PCA

#Scale the data
scaler = StandardScaler()
scaled_data = scaler.fit_transform(data)
#Apply PCA
pca = PCA()
principal_components = pca.fit_transform(scaled_data)

#Pairplot for 11 principle components
pc_df = pd.DataFrame(principal_components)
sns.pairplot(pc_df)

/Users/eren/micromamba/envs/Py_ML_DS_env/lib/python3.9/site-packages/seaborn/axisgrid.py:64: UserWarning: The figure layout has changed to tight
  self.fig.tight_layout(*args, **kwargs)

<seaborn.axisgrid.PairGrid at 0x2d3993d60>

#Explained variance ratio per principal component.
pca.explained_variance_ratio_

array([0.20715348, 0.11070825, 0.10003384, 0.09607237, 0.09187079,
       0.08845826, 0.08565904, 0.08165053, 0.07339863, 0.05830268,
       0.00669213])

# Cumulative sum of the explained variance ratio
cum_explained_variance= pca.explained_variance_ratio_.cumsum()
cum_explained_variance

array([0.20715348, 0.31786173, 0.41789557, 0.51396794, 0.60583873,
       0.69429699, 0.77995603, 0.86160656, 0.93500519, 0.99330787,
       1.        ])

pd.DataFrame({"Explained Variance Ratio": pca.explained_variance_ratio_, "Cumulative Explained Variance Ratio":cum_explained_variance}, index=[f"Comp.{i+1}" for i in (pc_df.columns)])

#Plot the cumulative sum of the explained variance ratio
plt.figure(figsize=(8, 6))
plt.plot(cum_explained_variance, marker='o', linestyle='--')
plt.title('Cumulative Explained Variance by PCA Components')
plt.xlabel('Number of Components')
plt.ylabel('Cumulative Explained Variance')
plt.figtext(0.5, -0.05, " Observe that, when we take only the first eight principal components we lose more than 10% of explained variance during dimensionality reduction.", 
            ha="center", fontsize=10, wrap=True)
plt.grid(True)
plt.show()

loadings_df = pd.DataFrame(pca.components_, columns=data.columns, index=[f"Comp.{i+1}" for i in (pc_df.columns)])
loadings_df

#Scale the data
scaler1 = StandardScaler()
scaled_data_s11 = scaler1.fit_transform(S11_target)
plt.figure(figsize=(8, 6))
#Apply PCA for s11
pca = PCA()
pca.fit(scaled_data_s11)
# Plot the explained variance ratio
plt.plot(pca.explained_variance_ratio_.cumsum(), label="S11_target_pca", c='g')
plt.title('Cumulative Explained Variance by PCA Components')
plt.xlabel('Number of Components')
plt.ylabel('Cumulative Explained Variance')
plt.grid(True)
plt.legend(loc="best", fontsize=10)
plt.show()

#Scale the data
scaler2 = StandardScaler()
scaled_data_real = scaler2.fit_transform(real_S11[1::].reset_index(drop=True))
#Apply PCA for real part
pca = PCA()
pca.fit(scaled_data_real)
# Plot the explained variance ratio
plt.figure(figsize=(8, 6))
plt.plot(pca.explained_variance_ratio_.cumsum() , label="real_pca", c='b')
plt.title('Cumulative Explained Variance by PCA Components')
plt.xlabel('Number of Components')
plt.ylabel('Cumulative Explained Variance')
plt.grid(True)
plt.legend(loc="best", fontsize=10)
plt.show()

#Scale the data
scaler3 = StandardScaler()
scaled_data_imag = scaler3.fit_transform(imag_S11[1::].reset_index(drop=True))
#Apply PCA for real part
pca = PCA()
pca.fit(scaled_data_imag)
# Plot the explained variance ratio
plt.figure(figsize=(8, 6))
plt.plot(pca.explained_variance_ratio_.cumsum() , label="imag_pca", c='r')
plt.title('Cumulative Explained Variance by PCA Components')
plt.xlabel('Number of Components')
plt.ylabel('Cumulative Explained Variance')
plt.grid(True)
plt.legend(loc="best", fontsize=10)
plt.show()

pd.DataFrame({"Explained Variance Ratio": pca.explained_variance_ratio_, "Cumulative Explained Variance Ratio":pca.explained_variance_ratio_.cumsum()}, index=[f"Comp.{i+1}" for i in (S11_target.columns)])

scaled_data_real

array([[ 1.06574433,  1.01453916,  0.95853213, ..., -1.03117649,
        -1.03066207, -1.03006856],
       [-0.64499146, -0.6398629 , -0.63457663, ..., -0.9168636 ,
        -0.91321435, -0.90942875],
       [ 1.09232153,  1.10378008,  1.11527939, ...,  1.19098874,
         1.14704934,  1.10111919],
       ...,
       [-1.0315917 , -1.03139119, -1.03112695, ..., -0.92996344,
        -0.9291446 , -0.92822175],
       [-0.66015982, -0.65469567, -0.64906876, ..., -1.05988994,
        -1.0601784 , -1.06021103],
       [ 1.11677049,  1.1082833 ,  1.1004233 , ...,  1.28610211,
         1.29831994,  1.31073178]])

plt.figure(figsize=(8, 6))
#Apply PCA.
pca1 = PCA(10)
s11_target_pca= pca1.fit(scaled_data_s11)
# Plot the explained variance ratio
plt.plot(s11_target_pca.explained_variance_ratio_.cumsum(), marker='o', linestyle='--', label="S11_target_pca", c="g")
#Apply PCA.
pca2 = PCA(10)
real_pca=pca2.fit(scaled_data_real)
# Plot the explained variance ratio
plt.plot(real_pca.explained_variance_ratio_.cumsum(), marker='o', linestyle='--', label="real_pca",c="b")
#Apply PCA.
pca3 = PCA(10)
imag_pca=pca3.fit(scaled_data_imag)
# Plot the explained variance ratio
plt.plot(imag_pca.explained_variance_ratio_.cumsum(), marker='o', linestyle='--', label="imag_pca",c="r")

plt.title('Cumulative Explained Variance by PCA Components')
plt.xlabel('Number of Components')
plt.ylabel('Cumulative Explained Variance')
plt.figtext(0.5, -0.05, " With only 10 components selected it almost covers all of the variance of the data.", 
            ha="center", fontsize=10, wrap=True)
plt.grid(True)
plt.legend(loc="best", fontsize=10)
plt.show()

# Get the principle components (transformed)
S11_target_pca = pca1.fit_transform(scaled_data_s11)
Real_pca = pca2.fit_transform(scaled_data_real)
Imag_pca = pca3.fit_transform(scaled_data_imag)

# Reconstruct the original data from the principle components
S11_reconstructed = scaler1.inverse_transform(pca1.inverse_transform(S11_target_pca))
real_reconstructed = scaler2.inverse_transform(pca2.inverse_transform(Real_pca))
imag_reconstructed = scaler3.inverse_transform(pca3.inverse_transform(Imag_pca))

fig, ax = plt.subplots(figsize=(10, 6))
# Customize the plot
ax.set_title('3th Design (Reconstructed From 10 Principle Components)')
ax.set_xlabel('Frequency Index')
ax.grid(True)
# Add a caption below the plot
plt.figtext(0.5, -0.05, "", 
            ha="center", fontsize=10, wrap=True)
min_value = np.min(S11_reconstructed[2, :])
min_index = np.argmin(S11_reconstructed[2, :])
plt.text(min_index, min_value, f'x ({min_index}, {min_value:.5f})', fontsize=8, ha='left')
plt.plot(S11_reconstructed[2, :], label="Magnitude", c='g', lw=2.5)
plt.plot(real_reconstructed[2, :], label="Real Part", c='r',ls=':', lw=1.5)
plt.plot(imag_reconstructed[2, :], label="Imaginary Part", c='b',ls=':',lw=1.5)

ax.legend(loc="best", fontsize=10)

<matplotlib.legend.Legend at 0x369261160>

plt.figure(figsize=(8, 6))
#Apply PCA to get at least 90% explained variance.
pca1 = PCA(0.9)
s11_target_pca= pca1.fit(scaled_data_s11)
# Plot the explained variance ratio
plt.plot(s11_target_pca.explained_variance_ratio_.cumsum(), marker='o', linestyle='--', label="S11_target_pca", c="g")
#Apply PCA.
pca2 = PCA(4)
real_pca=pca2.fit(scaled_data_real)
# Plot the explained variance ratio
plt.plot(real_pca.explained_variance_ratio_.cumsum(), marker='o', linestyle='--', label="real_pca",c="b")
#Apply PCA.
pca3 = PCA(4)
imag_pca=pca3.fit(scaled_data_imag)
# Plot the explained variance ratio
plt.plot(imag_pca.explained_variance_ratio_.cumsum(), marker='o', linestyle='--', label="imag_pca",c="r")

plt.title('Cumulative Explained Variance by PCA Components')
plt.xlabel('Number of Components')
plt.ylabel('Cumulative Explained Variance')
plt.figtext(0.5, -0.05, " With 4 components selected we lose less than 10% of explained variance during dimensionality reduction.", 
            ha="center", fontsize=10, wrap=True)
plt.grid(True)
plt.legend(loc="best", fontsize=10)
plt.show()

# Get the principle components (transformed)
S11_target_pca = pca1.fit_transform(scaled_data_s11)
Real_pca = pca2.fit_transform(scaled_data_real)
Imag_pca = pca3.fit_transform(scaled_data_imag)

# Reconstruct the original data from the principle components
S11_reconstructed = scaler1.inverse_transform(pca1.inverse_transform(S11_target_pca))
real_reconstructed = scaler2.inverse_transform(pca2.inverse_transform(Real_pca))
imag_reconstructed = scaler3.inverse_transform(pca3.inverse_transform(Imag_pca))

fig, ax = plt.subplots(figsize=(10, 6))
# Customize the plot
ax.set_title('3th Design (Reconstructed From 4 Principle Components)')
ax.set_xlabel('Frequency Index')
ax.grid(True)
# Add a caption below the plot
plt.figtext(0.5, -0.05, "", 
            ha="center", fontsize=10, wrap=True)
min_value = np.min(S11_reconstructed[2, :])
min_index = np.argmin(S11_reconstructed[2, :])
plt.text(min_index, min_value, f'x ({min_index}, {min_value:.5f})', fontsize=8, ha='left')
plt.plot(S11_reconstructed[2, :], label="Magnitude", c='g', lw=2.5)
plt.plot(real_reconstructed[2, :], label="Real Part", c='r',ls=':', lw=1.5)
plt.plot(imag_reconstructed[2, :], label="Imaginary Part", c='b',ls=':',lw=1.5)

ax.legend(loc="best", fontsize=10)

<matplotlib.legend.Legend at 0x367b28a90>

from sklearn.pipeline import Pipeline
from sklearn.model_selection import train_test_split
from sklearn.ensemble import RandomForestRegressor
from sklearn.multioutput import MultiOutputRegressor
from sklearn.model_selection import GridSearchCV
from sklearn.metrics import make_scorer, mean_squared_error, accuracy_score

# Build the pipeline
pipe = Pipeline([
        ('scaler', StandardScaler()),
        ('reducer', PCA(n_components=8)),
        ('regressor', MultiOutputRegressor(RandomForestRegressor(random_state=101)))])
# Split the data and S11_target into training and testing data
X_train, X_test, y_train_201, y_test_201 = train_test_split(data, S11_target, test_size=0.3, random_state=101)
# Fit the pipeline to the training data
pipe.fit(X_train, y_train_201)
# Make predictions on the test set
y_pred_201 = pipe.predict(X_test)
# Calculate the mean squared error
MSE_1 = mean_squared_error(y_test_201, y_pred_201, multioutput='uniform_average')
# Prints the accuracy score
print(pipe.score(X_test, y_test_201))
print(MSE_1)

0.5594020626180238
0.019159774874724738

# Build the pipeline
pipe = Pipeline([
        ('scaler', StandardScaler()),
        ('reducer', PCA(n_components=8)),
        ('regressor', MultiOutputRegressor(RandomForestRegressor(random_state=101)))])
# Split the data and S11 parameters into training and testing data, use the same model using two different target sets.
X_train, X_test, y_train_201_real, y_test_201_real = train_test_split(data, real_S11[1::].reset_index(drop=True), test_size=0.3, random_state=101)
X_train, X_test, y_train_201_imag, y_test_201_imag = train_test_split(data, imag_S11[1::].reset_index(drop=True), test_size=0.3, random_state=101)
# Fit the pipeline to the training data and get predictions for both real and imaginary parameters.
pipe.fit(X_train, y_train_201_real)
real_201_predict= pipe.predict(X_test)
pipe.fit(X_train, y_train_201_imag)
imag_201_predict= pipe.predict(X_test)
# Calculate the S11 values
S11_magnitude_predict = np.sqrt(real_201_predict**2 + imag_201_predict**2)
S11_magnitude_test = np.sqrt(y_test_201_real**2 + y_test_201_imag**2)
# Score the mse on the test set
MSE_2 = mean_squared_error(S11_magnitude_test, S11_magnitude_predict)
# Prints the mse
print(MSE_2)

0.04225208448301995

# Build the pipeline
pipe = Pipeline([
        ('scaler', StandardScaler()),
        ('reducer', PCA(n_components=8)),
        ('regressor', MultiOutputRegressor(RandomForestRegressor(random_state=101)))])
# Split the data and S11_target into training and testing data
X_train, X_test, y_train, y_test = train_test_split(data, S11_target.iloc[:, key_frequencies], test_size=0.3, random_state=101)
# Fit the pipeline to the training data
pipe.fit(X_train, y_train)
# Make predictions on the test set
y_pred = pipe.predict(X_test)
# Calculate the mean squared error
MSE_3 = mean_squared_error(y_test, y_pred, multioutput='uniform_average')
# Prints the accuracy score
print(MSE_3)

0.018801862109574585

# Build the pipeline
pipe = Pipeline([
        ('scaler', StandardScaler()),
        ('reducer', PCA(n_components=8)),
        ('regressor', MultiOutputRegressor(RandomForestRegressor(random_state=101)))])
# Split the data and S11 parameters into training and testing data, use the same model using two different target sets.
X_train, X_test, y_train1, y_test1 = train_test_split(data, real_S11[1::].reset_index(drop=True).iloc[:, key_frequencies], test_size=0.3, random_state=101)
X_train, X_test, y_train2, y_test2 = train_test_split(data, imag_S11[1::].reset_index(drop=True).iloc[:, key_frequencies], test_size=0.3, random_state=101)
# Fit the pipeline to the training data and get predictions for both real and imaginary parameters.
pipe.fit(X_train, y_train1)
real_S11_predict= pipe.predict(X_test)
pipe.fit(X_train, y_train2)
imag_S11_predict= pipe.predict(X_test)
# Calculate the S11 values
S11_magnitude_predict = np.sqrt(real_S11_predict**2 + imag_S11_predict**2)
S11_magnitude_test = np.sqrt(y_test1**2 + y_test2**2)
# Score the mse on the test set
MSE_4 = mean_squared_error(S11_magnitude_test, S11_magnitude_predict)
# Prints the mse
print(MSE_4)

0.04507268081460516

# Build the pipeline
pipe = Pipeline([
        ('scaler', StandardScaler()),
        ('reducer', PCA(n_components=8)),
        ('regressor', RandomForestRegressor(random_state=101))])
# Split the data and S11_target into training and testing data
X_train, X_test, y_train_pca, y_test_pca = train_test_split(data, S11_target_pca, test_size=0.3, random_state=101)
# Fit the pipeline to the training data
pipe.fit(X_train, y_train_pca)
# Make predictions on the test set
y_pred_pca = pipe.predict(X_test)
# Calculate the mean squared error
MSE_5 = mean_squared_error(y_test_pca, y_pred_pca)
# Prints the accuracy score
print(MSE_5)

15.214632867210318

# Reconstruct the original data from the principle components
S11_pred_reconstructed = scaler1.inverse_transform(pca1.inverse_transform(y_pred_pca))
S11_test_reconstructed = scaler1.inverse_transform(pca1.inverse_transform(y_test_pca))

# Calculate the mean squared error of prediction and the test magnitude (reconstructed from 4 principle components)
MSE_5_1 = mean_squared_error(S11_pred_reconstructed, S11_test_reconstructed)
# Prints the mse
print(MSE_5_1)

0.013517684318910475

# Calculate the mean squared error of prediction and the real magnitude
MSE_5_2 = mean_squared_error(S11_pred_reconstructed, y_test_201) 
# Prints the mse
print(MSE_5_2)

0.017027761884641314

fig, ax = plt.subplots(figsize=(10, 6))
# Customize the plot
ax.set_title('51th Design Prediction (Reconstructed From 4 Principle Components)')
ax.set_xlabel('Frequency Index')
ax.grid(True)
# Add a caption below the plot
plt.figtext(0.5, -0.05, "", 
            ha="center", fontsize=10, wrap=True)
min_value = np.min(S11_pred_reconstructed[4, :])
min_index = np.argmin(S11_pred_reconstructed[4, :])
plt.text(min_index, min_value, f'x ({min_index}, {min_value:.5f})', fontsize=8, ha='left')
plt.plot(S11_pred_reconstructed[4, :], label="Prediction", c='b', lw=2)

min_value = np.min(S11_test_reconstructed[4, :])
min_index = np.argmin(S11_test_reconstructed[4, :])
plt.text(min_index, min_value, f'x ({min_index}, {min_value:.5f})', fontsize=8, ha='left')
plt.plot(S11_test_reconstructed[4, :], label="Test Data (Reduced Form)",c='g',  ls=':', lw=1.5)

min_value = np.min(y_test_201.iloc[4, :])
min_index = np.argmin(y_test_201.iloc[4, :])
plt.text(min_index, min_value, f'x ({min_index}, {min_value:.5f})', fontsize=8, ha='left')
plt.plot(y_test_201.iloc[4, :], label="Actual Data", c='r', ls=':', lw=1.5)

ax.legend(loc="best", fontsize=10)

<matplotlib.legend.Legend at 0x367232340>

# Build the pipeline
pipe = Pipeline([
        ('scaler', StandardScaler()),
        ('reducer', PCA(n_components=8)),
        ('regressor', RandomForestRegressor(random_state=101))])
# Split the data and S11_target into training and testing data
X_train, X_test, y_train_pca_real, y_test_pca_real = train_test_split(data, Real_pca, test_size=0.3, random_state=101)
X_train, X_test, y_train_pca_imag, y_test_pca_imag = train_test_split(data, Imag_pca, test_size=0.3, random_state=101)
# Fit the pipeline to the training data
pipe.fit(X_train, y_train_pca_real)
# Make predictions on the test set
y_pred_pca_real = pipe.predict(X_test)
# Fit the pipeline to the training data
pipe.fit(X_train, y_train_pca_imag)
# Make predictions on the test set
y_pred_pca_imag = pipe.predict(X_test)

# Calculate the mean squared error
MSE_6 = mean_squared_error(y_test_pca_real, y_pred_pca_real, multioutput='uniform_average')
# Prints the accuracy score
print(MSE_6)

# Calculate the mean squared error
MSE_7 = mean_squared_error(y_test_pca_imag, y_pred_pca_imag, multioutput='uniform_average')
# Prints the accuracy score
print(MSE_7)

8.7410756222609
36.02693734160932

# Reconstruct the original data from the principle components
real_pred_reconstructed = scaler2.inverse_transform(pca2.inverse_transform(y_pred_pca_real))
real_test_reconstructed = scaler2.inverse_transform(pca2.inverse_transform(y_test_pca_real))
imag_pred_reconstructed = scaler3.inverse_transform(pca3.inverse_transform(y_pred_pca_imag))
imag_test_reconstructed = scaler3.inverse_transform(pca3.inverse_transform(y_test_pca_imag))

fig, ax = plt.subplots(figsize=(10, 6))
# Customize the plot
ax.set_title('51th Design Prediction (Reconstructed From 4 Principle Components)')
ax.set_xlabel('Frequency Index')
ax.grid(True)
# Add a caption below the plot
plt.figtext(0.5, -0.05, "", 
            ha="center", fontsize=10, wrap=True)

plt.plot(real_pred_reconstructed[4, :], label="Real Part Prediction",marker='s',markevery=10, c='b', lw=2)
plt.plot(imag_pred_reconstructed[4, :], label="Imaginary Part Prediction",marker='o',markevery=10, c='b',lw=2)

plt.plot(real_test_reconstructed[4, :], label="Real Part (Test)",marker='s',markevery=10,markersize=5, c='g',ls=':', lw=1.5)
plt.plot(imag_test_reconstructed[4, :], label="Imaginary Part (Test)",marker='o',markevery=10,markersize=5, c='g',ls=':',lw=1.5)

plt.plot(y_test_201_real.iloc[4, :], label="Real Part (Actual)",marker='s',markevery=10,markersize=5, c='r',ls=':', lw=1.5)
plt.plot(y_test_201_imag.iloc[4, :], label="Imaginary Part (Actual)",marker='o',markevery=10,markersize=5, c='r',ls=':',lw=1.5)

ax.legend(loc="best", fontsize=10)

<matplotlib.legend.Legend at 0x36727ff70>

# Calculate the S11 values
S11_magnitude_predict = np.sqrt(real_pred_reconstructed**2 + imag_pred_reconstructed**2)
S11_magnitude_test = np.sqrt(real_test_reconstructed**2 + imag_test_reconstructed**2)
S11_magnitude = np.sqrt(y_test_201_real**2 + y_test_201_imag**2)

# Score the mse on the test set
MSE_7_1 = mean_squared_error(S11_magnitude_test, S11_magnitude_predict)
# Prints the mse
print(MSE_7_1)

0.044323588402768226

# Score the mse on the actual set
MSE_7_2 = mean_squared_error(S11_magnitude, S11_magnitude_predict)
# Prints the mse
print(MSE_7_2)

0.035406445184256806

fig, ax = plt.subplots(figsize=(10, 6))
# Customize the plot
ax.set_title('51th Design Prediction (Reconstructed From 4 Principle Components)')
ax.set_xlabel('Frequency Index')
ax.grid(True)
# Add a caption below the plot
plt.figtext(0.5, -0.05, "", 
            ha="center", fontsize=10, wrap=True)
min_value = np.min(S11_magnitude_predict[4, :])
min_index = np.argmin(S11_magnitude_predict[4, :])
plt.text(min_index, min_value, f'x ({min_index}, {min_value:.5f})', fontsize=8, ha='left')
plt.plot(S11_magnitude_predict[4, :], label="S11 Prediction", c='b', lw=2)

min_value = np.min(S11_magnitude_test[4, :])
min_index = np.argmin(S11_magnitude_test[4, :])
plt.text(min_index, min_value, f'x ({min_index}, {min_value:.5f})', fontsize=8, ha='left')
plt.plot(S11_magnitude_test[4, :], label="S11 Test", c='g', ls=':', lw=1.5)

min_value = np.min(S11_magnitude.iloc[4, :])
min_index = np.argmin(S11_magnitude.iloc[4, :])
plt.text(min_index, min_value, f'x ({min_index}, {min_value:.5f})', fontsize=8, ha='left')
plt.plot(S11_magnitude.iloc[4, :], label="S11 Actual Magnitude", c='r', ls=':', lw=1.5)

ax.legend(loc="best", fontsize=10)

<matplotlib.legend.Legend at 0x361db3ee0>

	length of patch	width of patch	height of patch	height of substrate	height of solder resist layer	radius of the probe	c_pad	c_antipad	c_probe	dielectric constant of substrate	dielectric constant of solder resist layer
0	3.202024	2.288742	0.012514	0.139247	0.041757	0.028566	0.000549	0.032403	0.348140	3.735926	4.278575
1	2.107848	2.895504	0.037171	0.149492	0.056775	0.028930	0.005536	0.053647	0.326369	4.929862	4.876068
2	3.252113	4.818411	0.025432	0.578834	0.029972	0.030922	0.020274	0.049845	0.446639	4.772670	4.745106
3	4.161509	2.294309	0.011058	0.117266	0.093223	0.017604	0.001135	0.098610	0.055665	4.102438	3.755671
4	4.820912	2.948325	0.019658	0.163503	0.094337	0.025757	0.021725	0.072813	0.272282	2.531031	3.047553
...	...	...	...	...	...	...	...	...	...	...	...
380	3.950131	5.165856	0.034389	0.770787	0.095579	0.025591	0.011430	0.048502	0.241219	4.099788	4.635040
381	2.676373	2.826965	0.011377	0.106795	0.055925	0.042376	0.005102	0.060532	0.294000	4.758303	4.205687
382	4.406785	1.814310	0.011187	0.146086	0.049041	0.040626	0.022849	0.033737	0.225794	2.826602	4.908306
383	2.540721	3.310983	0.015000	0.135650	0.082388	0.043490	0.011130	0.033451	0.370502	3.303852	4.764701
384	2.425315	4.998480	0.029217	0.605634	0.028654	0.037827	0.005896	0.033075	0.215161	4.354493	2.676568

	length of patch	width of patch	height of patch	height of substrate	height of solder resist layer	radius of the probe	c_pad	c_antipad	c_probe	dielectric constant of substrate	dielectric constant of solder resist layer
count	385.000000	385.000000	385.000000	385.000000	385.000000	385.000000	385.000000	385.000000	385.000000	385.000000	385.000000
mean	3.569210	3.536983	0.024273	0.347643	0.060065	0.032198	0.012797	0.060648	0.245586	3.704384	3.521911
std	0.966173	1.182100	0.008800	0.272738	0.023670	0.010352	0.007111	0.021503	0.111245	0.853877	0.871233
min	1.805658	1.801273	0.010008	0.100321	0.020039	0.015012	0.000003	0.025292	0.050810	2.023380	2.001679
25%	2.755534	2.501163	0.016194	0.126901	0.038689	0.023389	0.006985	0.042011	0.148565	2.998152	2.783710
50%	3.637716	3.215396	0.024198	0.155254	0.060764	0.030979	0.012454	0.060532	0.245049	3.866295	3.480916
75%	4.369311	4.829731	0.031688	0.649324	0.080247	0.041819	0.019014	0.078227	0.340203	4.375551	4.278575
max	5.199919	5.198689	0.039843	0.799082	0.099728	0.049960	0.024996	0.099945	0.449599	4.999324	4.999950

	length of patch	width of patch	height of patch	height of substrate	height of solder resist layer	radius of the probe	c_pad	c_antipad	c_probe	dielectric constant of substrate	dielectric constant of solder resist layer
length of patch	1.000000	-0.114174	-0.026032	-0.064344	0.044502	-0.069905	-0.005560	-0.009627	0.081735	-0.037448	-0.014496
width of patch	-0.114174	1.000000	0.091726	0.923739	-0.021056	0.035675	-0.044198	-0.013358	0.029999	0.442193	0.038746
height of patch	-0.026032	0.091726	1.000000	0.082833	0.036045	0.030689	0.002422	0.037583	0.031592	-0.044318	-0.037769
height of substrate	-0.064344	0.923739	0.082833	1.000000	-0.011870	0.017410	-0.031571	-0.009432	0.044166	0.459847	0.002421
height of solder resist layer	0.044502	-0.021056	0.036045	-0.011870	1.000000	-0.012756	-0.018044	-0.017071	0.024842	-0.010967	-0.038405
radius of the probe	-0.069905	0.035675	0.030689	0.017410	-0.012756	1.000000	0.002906	-0.000523	0.048088	-0.025728	0.007524
c_pad	-0.005560	-0.044198	0.002422	-0.031571	-0.018044	0.002906	1.000000	0.067678	-0.015500	-0.049318	0.015284
c_antipad	-0.009627	-0.013358	0.037583	-0.009432	-0.017071	-0.000523	0.067678	1.000000	-0.132321	-0.016741	0.067821
c_probe	0.081735	0.029999	0.031592	0.044166	0.024842	0.048088	-0.015500	-0.132321	1.000000	0.015640	-0.029101
dielectric constant of substrate	-0.037448	0.442193	-0.044318	0.459847	-0.010967	-0.025728	-0.049318	-0.016741	0.015640	1.000000	0.060402
dielectric constant of solder resist layer	-0.014496	0.038746	-0.037769	0.002421	-0.038405	0.007524	0.015284	0.067821	-0.029101	0.060402	1.000000

	0	1	2	3	4	5	6	7	8	9	...	191	192	193	194	195	196	197	198	199	200
0	0.649471	0.608623	0.563770	0.515485	0.464365	0.411022	0.356056	0.300044	0.243525	0.186992	...	-0.895451	-0.894250	-0.893045	-0.891835	-0.890620	-0.889402	-0.888178	-0.886951	-0.885718	-0.884482
1	-0.684443	-0.680926	-0.677355	-0.673727	-0.670041	-0.666296	-0.662491	-0.658624	-0.654692	-0.650696	...	-0.835303	-0.832276	-0.829209	-0.826105	-0.822965	-0.819791	-0.816584	-0.813346	-0.810078	-0.806782
2	0.670194	0.678183	0.685886	0.693294	0.700399	0.707191	0.713661	0.719799	0.725595	0.731037	...	0.693171	0.675653	0.656992	0.637113	0.615936	0.593384	0.569383	0.543870	0.516796	0.488139
3	-0.856026	-0.854962	-0.853892	-0.852815	-0.851731	-0.850642	-0.849545	-0.848442	-0.847333	-0.846217	...	-0.115205	-0.096197	-0.076456	-0.055953	-0.034660	-0.012550	0.010402	0.034219	0.058921	0.084521
4	-0.952876	-0.951901	-0.950906	-0.949892	-0.948860	-0.947811	-0.946747	-0.945668	-0.944575	-0.943469	...	-0.692349	-0.690720	-0.689088	-0.687453	-0.685813	-0.684171	-0.682524	-0.680874	-0.679221	-0.677564
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
380	0.712664	0.703017	0.692973	0.682540	0.671727	0.660545	0.649007	0.637130	0.624930	0.612431	...	0.548479	0.560256	0.571626	0.582598	0.593180	0.603380	0.613208	0.622674	0.631789	0.640562
381	-0.304243	-0.267997	-0.229050	-0.187238	-0.142418	-0.094486	-0.043392	0.010828	0.068016	0.127852	...	-0.898600	-0.897598	-0.896590	-0.895579	-0.894562	-0.893540	-0.892514	-0.891483	-0.890447	-0.889406
382	-0.985887	-0.986109	-0.986290	-0.986434	-0.986540	-0.986612	-0.986650	-0.986656	-0.986631	-0.986577	...	-0.831635	-0.830253	-0.828863	-0.827464	-0.826056	-0.824640	-0.823215	-0.821781	-0.820338	-0.818886
383	-0.696270	-0.692488	-0.688645	-0.684740	-0.680771	-0.676737	-0.672637	-0.668469	-0.664232	-0.659923	...	-0.906488	-0.906808	-0.906959	-0.906948	-0.906784	-0.906473	-0.906022	-0.905439	-0.904728	-0.903896
384	0.689258	0.681693	0.674312	0.667233	0.660581	0.654481	0.649055	0.644419	0.640675	0.637908	...	0.536188	0.546594	0.556813	0.566843	0.576688	0.586347	0.595821	0.605112	0.614219	0.623143

	length of patch	width of patch	height of patch	height of substrate	height of solder resist layer	radius of the probe	c_pad	c_antipad	c_probe	dielectric constant of substrate	dielectric constant of solder resist layer	100
0	3.202024	2.288742	0.012514	0.139247	0.041757	0.028566	0.000549	0.032403	0.348140	3.735926	4.278575	-0.975209
1	2.107848	2.895504	0.037171	0.149492	0.056775	0.028930	0.005536	0.053647	0.326369	4.929862	4.876068	0.522707
2	3.252113	4.818411	0.025432	0.578834	0.029972	0.030922	0.020274	0.049845	0.446639	4.772670	4.745106	0.386045
3	4.161509	2.294309	0.011058	0.117266	0.093223	0.017604	0.001135	0.098610	0.055665	4.102438	3.755671	-0.702926
4	4.820912	2.948325	0.019658	0.163503	0.094337	0.025757	0.021725	0.072813	0.272282	2.531031	3.047553	-0.826891
...	...	...	...	...	...	...	...	...	...	...	...	...
380	3.950131	5.165856	0.034389	0.770787	0.095579	0.025591	0.011430	0.048502	0.241219	4.099788	4.635040	0.743482
381	2.676373	2.826965	0.011377	0.106795	0.055925	0.042376	0.005102	0.060532	0.294000	4.758303	4.205687	-0.970511
382	4.406785	1.814310	0.011187	0.146086	0.049041	0.040626	0.022849	0.033737	0.225794	2.826602	4.908306	-0.930577
383	2.540721	3.310983	0.015000	0.135650	0.082388	0.043490	0.011130	0.033451	0.370502	3.303852	4.764701	0.470467
384	2.425315	4.998480	0.029217	0.605634	0.028654	0.037827	0.005896	0.033075	0.215161	4.354493	2.676568	-0.155731

Data Analysis¶

Data Analysis on The Input Data¶

Data Analysis on The Target Data¶

Relationship With The Input Data¶

Resonance Indices for The Target Data¶

Resonance Values As The Target¶

PCA with Scikit-Learn¶

Principle Component Analysis on The Input Data¶

Principle Component Analysis on The Target Data¶

Regression¶

1. Multitarget Regression¶

2. Multitarget (At Resonance Frequencies) Regression¶

3. Multi Target Regression (PCA Applied)¶

	0	1	2	3	4	5	6	7	8	9	...	191	192	193	194	195	196	197	198	199	200
0	0.781778	0.783440	0.785795	0.788790	0.792359	0.796433	0.800936	0.805793	0.810930	0.816277	...	0.988520	0.988551	0.988581	0.988610	0.988638	0.988664	0.988690	0.988714	0.988737	0.988759
1	0.986860	0.986669	0.986470	0.986263	0.986048	0.985824	0.985592	0.985350	0.985100	0.984839	...	0.945061	0.945880	0.946669	0.947428	0.948158	0.948861	0.949537	0.950188	0.950814	0.951417
2	0.866883	0.865643	0.864258	0.862724	0.861039	0.859198	0.857199	0.855036	0.852705	0.850202	...	0.865665	0.861328	0.856551	0.851268	0.845404	0.838870	0.831564	0.823368	0.814148	0.803752
3	0.995069	0.995055	0.995041	0.995025	0.995009	0.994991	0.994973	0.994953	0.994933	0.994912	...	0.935682	0.932755	0.929637	0.926312	0.922765	0.918978	0.914932	0.910607	0.905982	0.901033
4	0.985009	0.985235	0.985447	0.985645	0.985831	0.986005	0.986169	0.986321	0.986464	0.986597	...	0.988472	0.988418	0.988361	0.988304	0.988244	0.988183	0.988121	0.988057	0.987991	0.987924
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
380	0.712967	0.703130	0.692989	0.682545	0.671797	0.660746	0.649395	0.637749	0.625814	0.613598	...	0.675654	0.683714	0.691363	0.698625	0.705520	0.712067	0.718286	0.724194	0.729808	0.735143
381	0.901571	0.894376	0.886417	0.877604	0.867842	0.857028	0.845054	0.831812	0.817200	0.801129	...	0.990712	0.990714	0.990716	0.990716	0.990715	0.990713	0.990710	0.990706	0.990701	0.990695
382	0.985925	0.986114	0.986294	0.986465	0.986629	0.986785	0.986934	0.987076	0.987212	0.987341	...	0.987993	0.987889	0.987783	0.987674	0.987561	0.987445	0.987326	0.987204	0.987078	0.986949
383	0.984282	0.984049	0.983807	0.983558	0.983300	0.983034	0.982758	0.982473	0.982179	0.981875	...	0.911483	0.913052	0.914566	0.916027	0.917439	0.918802	0.920118	0.921390	0.922618	0.923805
384	0.710091	0.701981	0.694257	0.687033	0.680424	0.674545	0.669505	0.665398	0.662302	0.660269	...	0.818936	0.821178	0.823357	0.825474	0.827530	0.829524	0.831458	0.833330	0.835142	0.836893

	Explained Variance Ratio	Cumulative Explained Variance Ratio
Comp.1	0.207153	0.207153
Comp.2	0.110708	0.317862
Comp.3	0.100034	0.417896
Comp.4	0.096072	0.513968
Comp.5	0.091871	0.605839
Comp.6	0.088458	0.694297
Comp.7	0.085659	0.779956
Comp.8	0.081651	0.861607
Comp.9	0.073399	0.935005
Comp.10	0.058303	0.993308
Comp.11	0.006692	1.000000

	length of patch	width of patch	height of patch	height of substrate	height of solder resist layer	radius of the probe	c_pad	c_antipad	c_probe	dielectric constant of substrate	dielectric constant of solder resist layer
Comp.1	-0.101327	0.624178	0.071054	0.624137	-0.021049	0.025997	-0.054228	-0.018720	0.039213	0.445100	0.038422
Comp.2	0.315409	-0.009718	0.055716	0.018824	0.266259	0.002815	-0.265990	-0.559499	0.564753	-0.012829	-0.350944
Comp.3	-0.369645	0.040766	0.620460	0.016441	0.086417	0.548381	0.144827	0.088323	0.093131	-0.256167	-0.255339
Comp.4	0.376883	0.039631	0.392974	0.076518	0.510652	-0.454397	0.177533	0.373926	-0.151004	-0.022492	-0.184975
Comp.5	0.409833	0.012169	0.017701	0.028012	-0.124629	0.218986	0.580059	0.127759	0.487544	0.012254	0.420314
Comp.6	0.073314	-0.032326	0.069714	-0.060565	0.459841	0.304908	-0.547579	0.172140	0.016285	0.027772	0.591708
Comp.7	-0.253331	0.000413	-0.490098	0.002837	0.647199	0.212779	0.429976	-0.138146	-0.102528	0.097290	-0.086813
Comp.8	0.454266	0.022090	-0.317943	0.059481	-0.110408	0.491478	-0.174386	0.415883	-0.129279	0.053226	-0.460296
Comp.9	-0.406238	-0.040157	-0.216559	-0.038186	0.040184	-0.240280	-0.140321	0.548589	0.618694	0.031927	-0.145645
Comp.10	-0.036934	-0.328516	0.243013	-0.305362	-0.027430	0.080764	0.049041	-0.000574	0.000912	0.849121	-0.087887
Comp.11	-0.036141	-0.704172	0.006071	0.708004	-0.004432	0.010532	-0.010954	-0.006460	-0.008289	-0.018678	0.028625

	Explained Variance Ratio	Cumulative Explained Variance Ratio
Comp.1	4.167438e-01	0.416744
Comp.2	2.269942e-01	0.643738
Comp.3	1.619304e-01	0.805668
Comp.4	1.030973e-01	0.908766
Comp.5	3.782842e-02	0.946594
...	...	...
Comp.197	1.933790e-33	1.000000
Comp.198	1.801411e-33	1.000000
Comp.199	1.772045e-33	1.000000
Comp.200	1.561350e-33	1.000000
Comp.201	1.053047e-33	1.000000